Q&A: Undefined versus Infinity

Cross-posted from here.

Question

Is “undefined” a real answer in math? Doesn’t an answer of “undefined” really just mean that division by zero is crashing the calculator, which cannot store an infinite value? If that’s true, then why do we say it’s “undefined” rather than saying it’s “infinity”?

Answer

Thanks for the opportunity to clear this up. Undefined is a real answer and it does not always mean infinity. I’ll give some examples to illustrate. In these examples, keep in mind that A/B represents the number that you multiply by B to get A.

0/0 is undefined because ANY number, when multiplied by 0, produces 0. The expression 0/0 is not defined to refer to any specific one of those numbers.

1/0 is undefined because NO number, when multiplied by 0, produces 1. There is no number that fits this criterion.

Sure, if you consider 1/something, where “something” is a tiny positive number, then 1/something approaches infinity as “something” approaches 0. But that’s not the same as 1/0. That’s the limit of 1/x as x approaches 0 from the positive side. Totally different thing.

And you get a totally different result, NEGATIVE infinity, if you take the limit of 1/x as x approaches 0 from the NEGATIVE side.

So even if you’re asking about the limit of 1/x as x approaches 0, the answer is undefined because you get different results depending on whether you’re approaching from the positive side or the negative side.